For any fundamental matrix \(U \in \mathrm {GL}_n(\mathbf {L})\) , an easy computation shows that
\(U^{-1}\sigma (U) \in \mathrm {GL}_{n}(C_{\mathbf {L}})= \mathrm {GL}_{n}(C_{\mathbf {k}})\)  for all \(\sigma \in G\) .
By [1]}, the faithful representation
\(G& \rightarrow & \mathrm {GL}_{n}(C_{\mathbf {k}}) \\\sigma & \mapsto & U^{-1}\sigma (U)\) 
